Simple Machines: How Does a Lever Work?

© Eugene Brennan

What Is a Lever?

A lever is a simple device that works by changing the input force needed to do work and move something. Some levers do this by magnifying the force applied to them. In this tutorial, we'll explore how levers are used in our everyday lives and then learn the simple physics of how they work.

Examples of levers. © Eugene Brennan

The Lever: One of the Six Classical Simple Machines

The lever is one of the six simple machines which were defined by Renaissance scientists hundreds of years ago.

Six Classical Simple Machines

• Lever
• Wheel
• Inclined plane
• Screw
• Wedge
• Pulley

You've used a lever in some shape or form without actually realizing it. So for instance scissors, nutcrackers, pliers, hedge shears, bolt cutters and lopping shears all use levers in their design. A prybar or crowbar is a lever also, and when you prise open the lid of a tin with the handle of a spoon, you're using "the law of the lever" to create a greater force. A long handle on a wrench provides more "leverage". A claw hammer also acts as a lever when pulling out nails. A see-saw and wheelbarrow are also levers.

What Is a Force?

To understand how a lever works, we first need to learn about forces. A force can be thought of as a "push" or "pull". A force is required for example to lift a weight or slide it on a surface.

Examples of Forces

• A forklift lifting a load
• Tension in a spring when you pull on it
• A magnet pulling a piece of iron
• Air in a balloon, football or tire, pushing outwards on its walls
• The force of gravity keeping things on the ground
• Air or water resisting the movement of a car, aircraft or ship. This is called drag.

An active force results in a reactive force, so for instance when you pull on a spring, this is the active force. The tension in the spring is the reactive force pulling back.

Different types of forces. © Eugene Brennan

What Does Mechanical Advantage Mean?

A simple machine can magnify a force. The degree to which the force is magnified is called the mechanical advantage. Levers are great because they can increase mechanical advantage and generate much larger forces. For example a hammer or crowbar can easily produce a ton of force for pulling out nails, lifting a rock or prising up boards.

What Are the Parts of a Lever?

In general, levers can be actual tools or components in a machine, but they also appear in our bodies and in nature. A lever is made up of several parts:

• Beam: The physical lever itself is made of materials such as wood, metal or plastic, bone in humans and animals etc., which can pivot or move on the fulcrum.
• Effort: The force that is exerted on the lever
• Fulcrum: The point at which a lever pivots or hinges
• Load: The object that is acted on by the lever.

Levers can increase a force. I.e they give a mechanical advantage. © Eugene Brennan

Using the handle of a spoon to open a tin. The spoon acts as a lever, creating a larger force to lift the lid. The fulcrum is the rim of the tin. © Eugene Brennan

Examples of Levers in Everyday Life

• Crowbars and prybars
• Pliers
• Scissors
• Bottle openers
• Bolt cutters
• Nut crackers
• Claw hammer
• Wheel barrow
• Parts of machines such as engines and production machines in factories
• Bones and joints in your body

From "The World of Wonder" a children's science periodical from the 1930s.

Three Classes of Levers

The class of a lever depends on the position of the effort, fulcrum and load.

First Class Lever

The effort is on one side of the lever and the load is on the other side. The fulcrum is in the middle. Moving the fulcrum closer to the load increases the mechanical advantage and increases the force on the load.

Examples: Scissors, pliers, hammer

Second Class Lever

The effort is on one side of the lever and the fulcrum is on the other side with the load between the effort and fulcrum. Keeping the effort in the same position and moving the load closer to the fulcrum, increases the force on the load.

Examples: nutcracker, wheelbarrow

Third Class Lever

The fulcrum is on one end of the lever, the load is on the other side and the effort is between the load and fulcrum. A third class lever has less of a mechanical advantage than the other two types because the distance from the load to the fulcrum is greater than the distance from the effort to the fulcrum.

Examples: human arm, broom, sporting equipment (e.g. baseball bat)

Three classes of levers. © Eugene Brennan

Examples of Levers

Bolt cutters. Annawaldl, public domain image via Pixabay.com

Using a crowbar as a lever to lift a heavy piece of stone.  Public domain image via Pixabay.com

Pliers and side cutters. © Eugene Brennan

An excavator (digger) has several connected levers on its boom. Hydraulic cylinders produce the force required to move the levers. Didgeman, public domain image via Pixabay.com

The Physics of How Levers Work

What Is the Moment of a Force?

To understand how levers work, we first need to understand the concept of moment of a force. The moment of a force about a point is the magnitude of the force multiplied by the perpendicular distance from the point, to the line of direction of the force. So in the diagram below, if the magnitude of the force is F and the distance is d, the the moment = Fd

Moment of a force. © Eugene Brennan

In the 2nd diagram below, two forces act on a lever. This is a schematic or diagram, but it symbolically represents any of the real life levers mentioned above.

The lever pivots at a point called a fulcrum represented by the black triangle (in real life, this could be the screw holding the two blades of a scissors together). A lever is said to be balanced when the lever doesn't rotate and everything is in equilibrium (e.g. two people of equal weight sitting on a see-saw, at equal distances from the pivot point).

Forces on a lever. © Eugene Brennan

In the diagram, a force F1 acts downward on the lever at a distance d1 from the fulcrum.

Another force F2 at distance d2 from the fulcrum acts downwards on the lever. When the lever is balanced, F2 balances the effects of F1 and the lever is stationary, i.e. there is no net turning force.

When balanced:

"The sum of the clockwise moments equals the sum of the counter-clockwise moments"

So for F1, the clockwise moment is F1d1

and for F2, the counter-clockwise moment is F2d2

So the clockwise moments = the anticlockwise moments

and

F1d1 = F2d2

Imagine if F1 is the active force and is known. F2 is unknown but must push down on the lever to balance it.

Dividing both sides of the equation by d2 and switching the left and right gives:

F2 = F1(d1/d2)

So F2 must have this value to balance the force F1 acting down on the right-hand side.

Since the lever is balanced, we can think of there being an equivalent force equal to F2 (and due to F1), shown in orange in the diagram below, pushing upwards on the left side of the lever.

If the distance d2 is a lot smaller than d1 (which would be the case with a crowbar or pliers), the term (d1/d2) in the equation above is greater than unity and F2 becomes greater than F1. (a long-handled crowbar can easily produce a ton of force).

This is intuitively correct since we know how a long crowbar can create a lot of force for lifting or prying things, or if you put your fingers between the jaws of a pliers and squeeze, you know all about it!

If F2 is removed and the lever becomes unbalanced, the upwards force due to the force F1 on the right is still F1(d1/d2). This force magnifying effect or mechanical advantage of a lever is one of the features that makes it so useful.

When the lever is balanced, the force F1 produces an equivalent force of magnitude F2 (shown in orange). This balances F2 (shown in blue) acting downwards. © Eugene Brennan

The Law of the Lever

We can summarise the above reasoning into a simple equation known as the law of the lever:

Mechanical advantage = F2/F1 = d1/d2

d1 is called the effort arm and d2 the load arm. If F1 is the effort and F2 is the load, then:

Interesting Fact: We Have Levers in Our Body!

Many of the bones in your body act as third class levers. For instance in your arm, the elbow is the pivot, the biceps muscle creates the effort acting on the forearm and the load is held by a hand. The small bones in the ear also form a lever system. These bones are the hammer, anvil and stirrup and act as levers to magnify sound coming from the eardrum.

The bones in our arms and other part of the body are third class levers. Original image without text, OpenStax College, CC BY SA 3.0 unported via Wikimedia Commons

What Is a Counterbalance Used For?

A counterbalance is a weight added to one end of a lever or other pivoting structure so that it becomes balanced (the turning moments clockwise and anti-clockwise are equalised). The weight of the counterbalance and its position relative to the pivot are set so that the lever can stay at any angle without turning. The advantage of a counterbalance is that a lever only has to be displaced and doesn't have to be physically lifted. So for instance a heavy road barrier could be raised by a human if it moves freely on its pivot. If there was no counterbalance, they would have to push down a lot harder on the barrier to lift the other end. Counterbalances are also used on tower cranes to balance the boom so that the crane doesn't topple over. Swing bridges use counterbalances to balance the weight of the swing section. Sometimes the counterbalancing force is provided by a spring instead of a weight. For instance springs are sometimes used on the deck of a lawn mower so a person doesn't have to lift the deck when adjusting the height. Also springs might be used on the lid of a home appliance such as a chest freezer to stop the lid falling down when raised.

A counterbalance used to balance a lever. These are often seen on road barriers where one end of the lever is much shorter than the other end. © Eugene Brennan

A tower crane. The counterbalance consists of a collection of concrete slabs mounted near the end of the boom. Conquip, public domain image via Pixabay.com

Counterbalance on a similar crane. User:HighContrast, CC 3.0 via Wikimedia Commons

Not All Levers Magnify Force: Levers Increasing Range of Motion

We discovered that many levers have a mechanical advantage and increase the force on a load when an input force is applied to the lever. This is really useful in a tool such as a wire snips, bolt cutter or gardening lopping shears for creating large forces that can shear through materials. However another function of a lever is to increase range of motion. In this case, the force on the load is less than the input force to the lever, but the lever produces a greater range of motion. An example is the biceps and forearm. The biceps muscle typically can move the hand times eight times further than the displacement of the point where the muscle attaches to the arm. The increased range of motion is achieved by positioning the effort closer to the fulcrum than the load.

The biceps and forearm form a 3rd class lever system that increase range of motion. Niwadare, CC BY-SA 4.0 international via Wikimedia Commons

References

Curley, R. (2017, June 26). Simple machines. Encyclopaedia Britannica.

Hannah, J. and Hillerr, M. J., (1971) Applied Mechanics (First metric ed. 1971) Pitman Books Ltd., London, England.

Disclaimer

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2018 Eugene Brennan

Another Faulty LED Street Lamp

From the DeadSure app for reporting faulty street lighting.

Too early to make a judgement, but something tells me these new LED street lights are less reliable than the older sodium ones. Which makes sense, considering sodium lamps are simpler and purely electrical/electromechanical, whereas LED lamps have more working parts— not moving parts, but semiconductors—that can fail. I reported this one that I noticed tonight because the upper path in the Valley Park seemed dark. There may be a couple more not lighting, further out the road closer to St. Brigid's Well.

LED lighting theoretically has a lifespan in the tens of thousands of hours (30,000 to 40,000 hours). However, in reality it can be less than this because the driver electronics fails.

57 Modes and Nothing Wrong

Bicycle rear light. © Eugene Brennan

Mystery light.

I've bashed it almost to the point of cracking the lens, put it in the freezer for half an hour while turned on and left a strong magnet in its vicinity and it still won't turn off. Yet when I'm cycling, after 100 m or so, it manages to turn itself off. And that keeps happening. The only thing I can think of is that deceleration in a certain direction is bending a spring in a clicky button. Or maybe it's condensation on the PCB. Or possibly constant vibration rather than large shocks.

Anyway, I hate the way they don't put proper switches on these things instead of buttons for changing modes that inevitably cause them to switch off on bad roads. It's difficult to get a decent backlight anywhere. I've tried eBay, Amazon and AliExpress.

 

Archimedes' Principle, Buoyancy Experiments and Flotation Force

Archimedes' principle. © Eugene Brennan

What is Buoyancy?

We've all experienced buoyancy. If you throw a football into water, it floats on the surface. In a swimming pool, you can float on your back and large ships float in the sea. Even things that sink such as stones, experience a buoyant force, but the force is insufficient to keep them afloat. Buoyancy is a force that pushes up on the underside of an object that's placed in a fluid. The fluid doesn't have to be water, it can be a gas such as air and helium balloons experience buoyancy, causing them to rise upwards.

Who Was Archimedes?

Archimedes of Syracuse was a Greek astronomer, scientist and mathematician who was born circa 287 BC. Amongst his many works as a great scientist of the classical period was laying the ground work for modern calculus as well as proving geometric theorems, working out approximations for pi and calculating the surface area and volumes of 3D solids.

What is Archimedes' Principle?

Archimedes' principle, also called the principle of Archimedes or the Archimedean principle, states that the upthrust or buoyant force on an object in a fluid is equal to the weight of the displaced fluid. Displaced means pushed out of the way, so for instance when you drop stones into a container of water, you displace the water and it rises in the container. A force can be though of as a push or pull. The fluid doesn't have to be water, it can be any other liquid or gas, e.g. air. For more detailed information on forces, see my physics tutorial:

Examples of Forces in Everyday Life and How They Affect Things 

Negative, Positive and Neutral Buoyancy

An object placed in a fluid such as water can do three things:

1. It can sink. We call this negative buoyancy.
2. It can float. We call this positive buoyancy. If we push the object below the surface of the water and let go, the positive buoyancy force pushes it back up again above the surface.
3. It can stay submerged below the surface and neither sink to the bottom nor float back to the surface. If it's moved to a different depth below the surface, it stays in that position. This is called neutral buoyancy.

Experiments to Understand the Archimedean Principle

Let's do some experiments to investigate and understand the principle of Archimedes.

Step 1. Weigh the Object

Imagine we have an object suspended from a weighing scales. For instance it could be an iron weight like the one in the diagram below. We're going to lower it into a tank of water filled to the brim, level with the overflow outlet. The weight may float or it may become submerged, but it doesn't matter and doesn't affect our experiment. Before we lower it into the tank, the weighing scales tells us that its weight is 6 kg.

Experiment to investigate the principle of Archimedes. © Eugene Brennan

Step 2. Weigh the Displaced Water

As the weight is lowered, water is displaced and overflows into the pan on the second scales. When the weight is fully submerged we find that the water we collected weighs 2 kg.

Demonstrating the Archimedes' principle. Weight submerged in water. Displaced water is weighed. © Eugene Brennan

Step 3. Check the Weight on the First Scales

We now check the weight on the first scales again.

We find that the weight indicated is only 4 kg this time.

Step 4. Do Some Calculations

We discover that when we subtract the new measurement of the iron weight indicated on the scales from the previous measurement, it tallies with the weight we measured on the second scales.

So 6 kg - 4 kg = 2 kg

Principle of Archimedes

We've just discovered Archimedes' principle!

"The upthrust on a body submerged or floating in a fluid equals the weight of the fluid displaced"

How come the weight indicated on the first scales is now less than it was before?

It's because of the upthrust or buoyancy force.

This accounts for the difference and the object appearing lighter.

The 6 kg weight acts downwards, but it's as though 2 kg is pushing upwards acting as support and lessening the weight of the iron. So the scales indicates a smaller net weight of 4 kg. This upthrust equals the weight of the displaced water we collected in the pan of the second scales.

However, the mass of the object is still the same = 6 kg. The mass of objects stays the same, assuming they don't lose any material. Weight however changes. Weight is the force on an object due to gravity. On the Moon, where the force is smaller, objects weigh less.

The principle of Archimedes. Buoyant force equals the weight of the displaced liquid. © Eugene Brennan

Negative Buoyancy and Sinking Bodies

In the first experiment we did earlier, the iron weight sank below the water as it was lowered. The 6 kg iron weight we used displaces water. However the weight of the water displaced is only 2 kg. So according to the principle of Archimedes, the buoyant force is 2 kg acting upwards on the iron weight. Since this is less than 6 kg, it isn't enough to support the weight in the water. We call this negative buoyancy. If the weight was detached from the hook of the weighing scales, it would sink.

Negative buoyancy. Buoyant force is less than the weight of the submerged body. © Eugene Brennan

What are Examples of Things That Need Negative Buoyancy?

• Anchors need to have negative buoyancy so they can sink to the ocean floor.
• Fishing net sinkers to keep nets open

An anchor on a ship. Analogicus via Pixabay.com

Experiment 2. Investigating Positive Buoyancy

This time we lower a hollow steel ball onto the surface.

Positive Buoyancy and Floating Objects

What happens if a weight floats and doesn't sink? In the diagram below we lower a hollow steel ball into the tank. This time we know the weight is 3 kg. The chain goes slack if we move the scales closer to the water surface, because the weight floats and doesn't pull down on it. (In reality the chain has weight and will pull down on the scales, but for the sake of the experiment, let's imagine it's weightless.) The scale indicates 0 kg. The water displaced weighs the same as the weight this time.

So what happens in this scenario is that the ball displaces water and settles lower and lower in the water until the upthrust equals its weight. The force of gravity on the object acting downwards, i.e. its weight, is balanced by a buoyant force or upthrust acting upwards. Since the two are the same, the object floats.

In this second scenario, the object doesn't become fully submerged.

If we push the ball below the surface, it will displace more water, increasing the buoyant force. This force will be greater than the weight of the ball and the positive buoyancy will cause it to rise up out of the water and just displace enough water until the buoyant force and weight are equal again.

Positive buoyancy. The buoyant force and weight of the hollow steel ball are equal. © Eugene Brennan

 

What are Examples of Things That Need Positive Buoyancy?

• Lifebelts (lifebuoys)
• Marking and meteorological buoys
• Ships
• Swimmers
• Life jackets
• Floats on fishing lines
• Floats in toilet cisterns and float switches
• Flotation tanks/bags for recovering lost cargo/archaeological artefacts/submerged vessels
• Floating oil rigs and wind turbines

Things that need to have positive buoyancy. Clockwise from the top: A life belt, marking buoy, swimmer, ship.

Experiment 3. Investigating Neutral Buoyancy

In this experiment, the object we use has neutral buoyancy and can stay suspended under the water surface without sinking down or being pushed back up by the buoyant force of the water.

Neutral buoyancy occurs when the average density of an object is the same as the density of the fluid it is immersed in. When the object is below the surface, it neither sinks nor floats. It can be positioned at any depth below the surface and will stay there until another force moves it to a new location.

Neutral Buoyancy. Body can be positioned anywhere under the surface. Buoyancy force and weight of ball are equal. © Eugene Brennan

What are Examples of Things That Need Neutral Buoyancy?

• Diver
• Submarine

Submarines need to be able to control their buoyancy. So when there is a requirement to dive, large tanks are filled with water, producing negative buoyancy enabling them to sink. Once they reach the required depth, buoyancy is stabilised so that it becomes neutral. The sub can then cruise at a constant depth. When the sub needs to rise again, water is pumped out of the ballast tanks and replaced by air from compression tanks. This gives the submarine positive buoyancy, allowing it to float to the surface.

Humans naturally float in a vertical position with their noses just under the water if they relax their muscles. Scuba divers keep their buoyancy neutral by using belts with lead weights attached. This allows them to stay underwater at a desired depth without having to continually swim downwards.

A scuba diver needs to have neutral buoyancy. A submarine needs to have neutral, positive and negative buoyancy. Skeeze and Joakant. Public domain images via Pixabay.com

Negative, neutral and positive buoyancy. 

Formula for the Buoyant force

 

Where

ρ is the density of the displaced fluid

V is the volume of the displaced fluid

and g is the acceleration due to gravity

If we choose the the weight acting downwards to be a positive force, the negative sign in the equation for buoyant force is because it is a vector and acts in the opposite direction.

The weight of an object is

F_g = mg

where m is the mass of the object

Why do Ships Float?

Ships weigh thousands of tons, so how come they can float? If a stone or a coin is dropped into water it will sink straight to the bottom.

The reason ships float is because they displace lots of water. Think of all the space inside a ship. When a ship is launched into water, it pushes all the water out of the way and the massive upthrust balances the downwards weight of the ship, allowing it to float.

Why do Ships Sink?

Positive buoyancy keeps a ship afloat because the weight of the ship and buoyant force are balanced. However if too much heavy cargo is taken on by a ship, its total weight could exceed the buoyant force and it could sink. If the hull of a ship is holed, water will run into the hold. As water rises in the ship, it weighs down on the inside of the hull, causing the total weight to be greater than the buoyant force, making the ship sink.

A ship would also sink if we could magically crush all the steel structures and hull into a block. Because the block would take up a small fraction of the original volume of the ship, it wouldn't have the same displacement and therefore negative buoyancy.

Ships float because they displace a huge amount of water and the buoyant force can support the weight of the ship. Susannp4, public domain image via Pixabay.com

How Does Density of a Liquid Affect Buoyancy?

The density of the fluid an object is placed in affects buoyancy, however Archimedes' principle still applies.

Average density of object

If m is the mass of an object and V is its volume, then the average density ρ of the object is:

ρ = m / V

An object may not be homogenous. This means that the density could vary throughout the volume of the object. For instance if we have a large, hollow steel ball, the density of the steel shell would be about 8000 times the density of the air inside it. The ball could weigh tons, however when we work out the average density using the equation above, if the diameter is large, the average density is much less than the density of a solid steel ball because the mass is a lot less. If the density is less than that of water, the ball will float when placed in water.

Buoyancy and average density

• If the average density of an object is > density of the fluid, it will have negative buoyancy
• If the average density of an object is < density of the fluid, it will have positive buoyancy
• If the average density of an object = density of the fluid, it will have neutral buoyancy

Remember for an object to float, its average density must be lower than the density of the fluid it is placed in. So for instance if the density is less than water but greater than that of kerosene, it will float in water, but not in kerosene. 

A coin floats in mercury because the density of the metal that the coin is made from is less than that of mercury. Alby, CC BY-SA 3.0 via Wikimedia Commons

How Do Helium Balloons Float?

The principle of Archimedes works for objects not just in a liquid like water, but other fluids also, like air. Just like an aeroplane, a balloon needs a force called lift to make it rise in the air. Balloons don't have wings to provide lift and instead use the buoyant force of displaced air.

Hot air and helium balloons rely on buoyancy to give them lift and keep them aloft. © Eugene Brennan

What gives a balloon lift to rise in the surrounding air?

Remember the Archimedes principle states that the upthrust or buoyant force is equal to the weight of the displaced fluid. In the case of a balloon, the displaced fluid is air.

First let's imagine a scenario where we have a large balloon and just fill it with air. The weight acting downwards consists of the weight of the balloon plus the weight of the air inside. However the buoyancy force is the weight of the displaced air (which is approximately the same as the weight of the air inside the balloon, because the displaced air has the same volume, neglecting the volume of the balloon material).

So the force acting downwards = weight of balloon + weight of air inside balloon

From Archimedes' Principle, the force acting upwards = weight of displaced air ≈ weight of air inside balloon

Net force acting downwards = (weight of balloon + weight of air inside balloon) - weight of air inside balloon = weight of balloon

Therefore the balloon will sink.

Weight of balloon and air inside (and also the basket and people, ropes etc) is greater than the buoyant force which is the weight of displaced air, so it sinks. © Eugene Brennan

Now imagine we make the balloon large so that it has a lot of space inside.

Let's make it a sphere 10 metres in diameter and fill it with helium. Helium has a density less than that of air.

The volume is approximately 524 cubic metres.

This much helium weighs about 94 kilos.

The balloon displaces 524 cubic metres of air, however air is nearly six times denser than helium, so that air weighs about 642 kg.

So from Archimedes principle, we know that the upthrust equals this weight. The upthrust of 642 kg acting upwards on the balloon is greater than the weight of the helium inside the balloon and this gives it lift.

In practice the weight acting downwards would be greater because of the weight of the skin of the balloon, basket, people etc.

Weight of balloon and helium inside it is less than weight of displaced air, so the buoyant force gives enough lift to make it rise. © Eugene Brennan

Why Do Hot Air Balloons Float ?

Helium balloons float because they're filled with helium which is less dense than air. Hot air balloons have tanks of propane and burners on board in the basket. Propane is the gas used for camping stoves and outdoor cooking grills. When the gas is burned, it heats the air. This rises upwards and fills the balloon, displacing the air inside. Because the air inside the balloon is hotter than the ambient temperature of the air outside, it's less dense and weighs less. So the air displaced by the balloon is heavier than the air inside it. Since the upthrust force equals the weight of the displaced air, this exceeds the weight of the balloon and the less dense hot air inside it and this lift force causes the balloon to rise.

A hot air balloon. Stux, public domain image via Pixabay.com

The weight of displaced air (which produces the buoyant force) is greater than the weight of the balloon's skin, basket, burners and less dense hot air inside it and this gives it enough lift to rise. © Eugene Brennan

Worked Examples on Buoyancy

Question 1:

A hollow steel ball weighing 10 kg and diameter 30 cm is pushed below the surface of water in a pool.
Calculate the net force pushing the ball back to the surface.

Calculate the buoyant force on a steel ball submerged in water. © Eugene Brennan

Answer:

We need to calculate the volume of water displaced. Then knowing the density of water, we can work out the weight of water and thus the buoyant force.

Volume of a sphere V = 4/3 π r³

r is the radius of of the sphere

π = 3.1416 approx

We know the diameter of the sphere is 30 cm = 30 x 10^-2 m

so r = 15 x 10^-2 m

Substituting for r and π gives us

V = 4/3 x 3.1416 x (15 x 10^-2)³

Now work out the mass of water displaced by this volume.

ρ = m / V

where ρ is the density of a material, m is its mass and V is the volume.

Rearranging

m = ρV

for pure water ρ = 1000 kg / m³

Substituting for ρ and V calculated previously gives us the mass m

m = ρV = 1000 x 4/3 x 3.1416 x (15 x 10^-2)³

= 14.137 kg approx

So the ball weighs 10 kg, but the displaced water weighs 14.137 kg. This results in a buoyant force of 14.137 kg acting upwards.

The net force pushing the ball to the surface is 14.137 - 10 = 4.137 kg

The ball has positive buoyancy, so it will rise to the surface and float, stabilizing with enough of its volume submerged to displace 10kg of water to balance its own 10kg weight.

Note: Strictly speaking force is measured in Newtons. If m is mass and g is the acceleration due to gravity = 9.81 approx, then force F = mg. A weight of 1 kg is equivalent to approximately 1 x g = 1 x 9.81 = 9.81 N
See my guide on mechanics for more info:

Newton's Laws of Motion and Understanding Force, Mass, Acceleration, Velocity, Friction, Power and Vectors

Question 2:

How many balloons would it take to lift a person if the balloons were filled with helium? Each balloon is spherical, 30 cm diameter and weighs 10g (including string). The person weighs 90 kg.
Assume that the thickness of the skin of the balloon is small compared to the diameter, so the volume of helium internally = volume of air displaced.

How many helium balloons does it take to lift a person? © Eugene Brennan

 

Answer:

First assign variables and convert to standard SI units.

Let d be the diameter of a balloon = 30 cm = 30 x 10^-2 m

so r the radius of a balloon = 15 x 10^-2 m

Let N be the number of balloons.

Let ρ_h be the density of helium = 0.1786 g / L = 1.786 x 10^-1 kg/m³

Let ρ_a be the density of air = 1.225 kg/m³

Let V be the volume of a balloon

Let w be the weight of each balloon = 10 g = 10 x 10^-3 kg

Let W_p be the weight of the person = 90 kg

The volume of each balloon is V = 4/3πr³

Total volume of all the balloons = NV

Total weight of helium = volume of balloons x density of helium = NVρ_h

Total weight of air displaced = buoyancy force = volume of balloons x density of air = NVρ_a

Total weight of balloons = Nw

Buoyancy force = weight of air displaced and this must be greater than the weight of the helium + the weight of balloons + person's weight.

Volume of a balloon V = 4/3πr³ = 4/3 x 3.1416 x (15 x 10^-2)³ = 1.414 x 10^-2 m³

So write the equation:

NVρ_a > NVρ_h + Nw + W_p

Rearrange giving N(Vρ_a - Vρ_h - w) > W_p

So N > W_p / (Vρ_a - Vρ_h - w)

Plug in the numbers:

N > 90 / (1.414 x 10^-2 x 1.225 - 1.414 x 10^-2 x 1.786 x 10^-1 - 10 x 10^-3) = 18,765

That's a lot of balloons and the problem is that as the number of the balloons increases, the helium also has to lift the weight of the material they're made of. A better solution is to use bigger balloons. If we increase the diameter, volume increases with the cube of the radius while surface area only increases with the square of the radius. So there's more volume of helium per weight of balloon and less balloons are needed. Imagine we increase the size of the balloons by a factor of ten from 30 cm to 3 m.

Surface area = 4πr²

If r is now 10 times greater, the r squared factor results in a 100 times increase in surface area (assuming the material is stretched by the same amount on inflation.) So weight increases 100 times and a balloon now weighs 100 x 10 g = 1 kg.

However volume is proportional to r³ so if r increases ten times, volume increases 1000 times, but balloon weight has only increased 100 times. The bottom line is that a bigger balloon has greater net lifting force, not just because of the bigger dimensions, but because balloon weight/volume becomes smaller.

The volume of a balloon is now:

V = 4/3πr³ = 4/3 x 3.1416 x (1.5 )³ = 14.137 m³

Plug in the numbers again:

N > 90 / (14.137 x 1.225 - 14.137x 1.786 x 10^-1 - 1) = 6.5

That's a bit better, so only 7 balloons needed this time.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional. 

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© 2019 Eugene Brennan